Disruption of the vortex-wave interaction self-sustaining process in stratified plane Couette flow

Minimal seeds for turbulence, initial conditions of smallest possible energy density $E_0=E_c$ that eventually transition to turbulence, closely follow the edge manifold in state space before leaving the edge manifold for the turbulent attractor. The trajectories visit a number of coherent states, exact solutions to the Navier--Stokes equations, that are embedded within the edge manifold. In unstratified plane Couette flow these `edge states' are manifestations of the `self-sustaining process' (SSP) of Waleffe (1997) or the `vortex-wave interaction' (VWI) of Hall and Smith (1991). We show that in density stratified plane Couette flow where both a constant, statically stable density difference $2\Delta\rho$ and a constant velocity difference $2\Delta U$ is maintained across a channel of depth $2h$, these states differ from the unstratified states at very small bulk Richardson numbers $Ri_B=g\Delta\rho h/\rho_0\Delta U^2$ (where $g$ is the gravitational acceleration and $\rho_0\gg\Delta\rho$ is a reference density) and that the new states are not of SSP/VWI type. We present a scaling argument to show this is to be expected for $Ri_B\geq O(1/Re)$, where $Re=\Delta Uh/\nu$ and $\nu$ is the kinematic viscosity, and investigate the mechanisms through which the SSP/VWI states breakdown.